Phased arrays provide new possibilities for remote sensing with radars. By imposing an azimuthal phase variation, electromagnetic beams that carry orbital angular momentum can be formed. Such beams have a phase structure that appears twisted and as a result an intensity null in the center of the beam cross section. Here we numerically investigate twisted beams for incoherent scatter radars that are used to study the ionosphere. We discuss the possibility of utilizing such radar beams to probe twisted beams of plasma waves and flows transverse to the beam axis, such as associated with auroral arcs. Transverse plasma flows may give rise to a rotational frequency shift of the scatter from a twisted beam and Doppler broadening due to the beam divergence, the latter also occurring with regular beams. Although the angular momentum effects of the considered large scale flows are generally small, sheared and vortical flows transverse to the beam axis can in principle be discriminated from unidirectional flows with beams carrying orbital angular momentum.